We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only $|X(k-1)|=O(n)$ points in contrast to $\binom{n}{k}$ points in the $k$-slice (which consists of all $n$-bit strings with exactly $k$ ones).
翻译:我们开始对高维扩张器进行布林功能分析。 我们给高维扩张的随机行走定义, 与早期的双向链接扩展器定义相吻合。 我们用这个定义描述一个模拟的Fourier扩张和Fourier 超立方体的Fleier 的模拟。 我们的模拟是分解到与模拟复合体相关的随机行走的大约异形空间。 我们的随机行走定义和分解具有额外优势, 包括高维扩张器和格拉斯曼摆放的早期定义。 我们用这个定义描述一个模拟Fleier 扩张器的Fleier 扩张器和 Fleier 超立方立方体的Fleier 。 我们的模拟显示, 恒定度高维扩张器有时可以作为Boolean 切片或超立方块的稀薄模型, 以及布利安功能分析的相当可能额外的结果, 包括高维度扩张器的扩张器, 包括高维特的扩张器和格拉斯曼摆的扩张器。 因此, 我们用这个解式的模型, 可以将这个模型, 美元 的比勒· 美元 的比克 的比勒 的比 美元 的比勒 的比 的比 。